Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics

Waves in the Ocean and Atmosphere presents a study of the fundamental theory of waves appropriate for first year graduate students in oceanography, meteorology and associated sciences. Starting with an elementary overview of the basic wave concept, specific wave phenomena are then examined, including: surface gravity waves, internal gravity waves, lee waves, waves in the presence of rotation, geostrophic adjustment, quasi-geostrophic waves and potential vorticity, wave-mean flow interaction and unstable waves. Each wave topic is used to introduce either a new technique or concept in general wave theory. Emphasis is placed on connectivity between the various subjects and on the physical interpretation of the mathematical results. The book contains numerous exercises at the end of the respective chapters.

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this implies that we need co » Q. - To ignore friction, compare d/dt with /iA:2,where A: is a typical value of wave- number—> cay> jik2. Fig. 3.1. The homogeneous layer of constant fluid supporting surface gravity waves ^_^^____.^__-__ D
Lecture 3 • Equations of Motion; Surface Gravity Waves 21 - To ignore nonlinearity, compare d/dt with respect to ii • V > co » uk or c » u <=: this is the condition that the disturbance be wave-like, i.e., that the signal is carried by the wave rather than the advective motion of the fluid. 2. Can we treat the fluid as incompressible? Assume we can linearize. Suppose the motion is adiabatic. In general then, we have ^• = 0, s = s{p,p) C.7) at with the linearization *=0=*&+*&. C.8) dt dp dt dp dt Thus, dp _ ds/dp dp _( dp] dp C.9) dt ds/dp dt {dp)sdt From the theory of acoustics we know (or we can easily find out) that the speed of sound in any medium is in fact given by the adiabatic compressibility of the medium. That implies that if ca is the speed of sound in a fluid, (One of the few scientific mistakes Newton made was to imagine that the speed of sound was this derivative at constant temperature and not entropy). So we have the estimate for the relation between a perturbation in the density and the perturbation of the pressure: Sp = clSp C.10) We can, on the other hand, estimate the magnitude of the pressure fluctuation from the horizontal momentum equation; if ~^j then from which it follows from the relation between the pressure and density disturbances: Hz)
22 Lecture 3 • Equations of Motion; Surface Gravity Waves Thus, 1 dSp _ | UOJ ~~p~lH~ [c*k We should compare this term, which is the estimate of the size of the local time derivative in the mass conservation equation with a typical term in the remaining combination of terms, namely, V • u = O(ku). Their ratio is thus dSp V-5 Thus, as long as the phase speed of the wave is small compared to the speed of sound, we can approximate the wave motion occurring as in an incompressible fluid for which the equation for mass conservation reduces to the condition V-m = 0 C.13) Note again that this does not by itself imply that dp/ dt = 0. A separate consider- consideration of the thermodynamics and the strength of the dissipation is required for that. We now have a series of parameter tests we can make after the fact to check to see whether the approximations of 1. linear motion 2. inviscid motion 3. incompressible motion 4. nonrotating dynamics will be valid. Assuming that these conditions will be met by the waves under consideration here, the equations of motion reduce to the much simpler set: p^-=-Vp-pgz C.14a) dt V-m = 0 C.14b) where z is a unit vector in the direction antiparallel to the direction of the local gravi- gravitation. We could have just waved our hands (perhaps appropriately for a course on waves) and written down these traditional approximate equations. However, it is impor- important for each new investigation of a wave type to carefully consider a priori the condi- conditions required to achieve the approximate dynamics used for the physical description of the wave to make sure that our physical system is no more complicated than it need be, while at the same time, it should be consistent with the underlying physics of the fluid.
Lecture 3 • Equations of Motion; Surface Gravity Waves 23 The curl of our momentum equation (recall that we are considering a fluid of con- constant density; the student is invited to use the thermodynamic equation to find the condition for the validity of that approximation) yields 0 C.15) dt So, if the vorticity is zero initially or at any instant (as it would be for an oscillatory motion for which each field goes through zero periodically), it follows that it remains zero for all time. If the curl of the velocity is zero, it follows from a fundamental fact of vector calculus that the velocity can be represented by a velocity potential, 0, w = V0 C.16) Note that only the spatial gradients of the velocity potential carry physical infor- information. Any arbitrary function of time can be added to 0 without changing the veloc- velocity field. Since the motion is incompressible, V.w = V-(V0)=V20 = O C.17) The equation of motion within the fluid thus reduces to the elliptic problem gov- governed by Laplace's equation: V20 = O C.18) This is an amazing simplification, and it should be a little disconcerting, because we are looking to describe a wave motion. Laplace's equation, by itself, is certainly not a wave equation. It describes among other things the electrical potential of static charges as well as certain static gravitational fields but, alone, no dynamical wave mechanism. The resolution of this seeming paradox is of course connected to the fact that we have not yet considered the boundary conditions for our problem. There is no more illuminating example of the importance of boundary conditions in the specifi- specification of the problem than this case of surface gravity waves. All the dynamics are in the boundary conditions. The internal equation, i.e., Laplace's equation merely relates the horizontal and vertical structure of the motion field. Boundary Conditions The obvious boundary condition at the lower horizontal surface is that the normal velocity vanishes there, i.e., w = 0 at z = -D, or ^- = 0, z = -D C.19) The boundary conditions at the upper surface are significantly more interesting. Let's call the departure of the free surface from its level "rest position" 77 (*,;>, t) (Fig. 3.2),
24 Lecture 3 • Equations of Motion; Surface Gravity Waves z = r\{x,y,t) P = Paix.y.t) Fig. 3.2. A definition figure for variables describing the motion in the surface gravity wave which must be small (this will presently be made more explicit). Thus, we consider the rippled free surface to only be slightly in departure from its rest state. At the free surface, the physical boundary conditions are 1. the dynamic condition: x,y&t) = p&(x,y,t) z=r\ C.20) and 2. the kinematic condition: _d(/> _dr/ dr/ dz dt dt C.21a) or dn C.21b) We must now write these conditions completely in terms of the velocity potential, 0. The linearized momentum equation is du dVd Vp _ — = —— = ——-qS/z dt dt p 6 or C.22a) C.22b) The integral of the last equation implies that C.23) where F(t) is an arbitrary function only of time. We can always add a function that is only of time to the velocity potential without changing the physical meaning of that potential. Let's imagine that we have added such an additional term such that its de- derivative with respect to time is equal to F(t). This allows us to write this linearized form of Bernoulli's equation everywhere in the fluid in the form:
Lecture 3 • Equations of Motion; Surface Gravity Waves 25 g 0 C.24) dt p Now let's apply this equation to the upper surface where z = T](x,y,t) andp = pa(x,y,t). Thus, g] o C.25) dt p A derivative of this equation with respect to time yields, using the kinematic con- condition on the upper surface, dt2 dz p dt Note that each term in this boundary condition is linear. The condition is applied at the unknown location z = t]. Indeed, the position of the free surface is, after all, one of the principal unknowns of the problem that we are try- trying to predict. For the general nonlinear problem, this unknown location of the bound- boundary, at which the important boundary conditions are applied, is one of the most diffi- difficult aspects of the problem. However, we are considering only the linear small ampli- amplitude problem, and it turns out that we can apply the boundary condition at the original position of the interface, i.e., at z = 0. To see this take any term on the left-hand side of the above boundary condition, generically call it G(x,jv,r]) and expand it around r\ = 0. Thus, r)C + higher order terms C.27) z=0 The first term on the right-hand side of the above equation is of the order of the amplitude of the motion, since G is one of the dynamic variables. Note that all the dynamical variables are linear in the size of the amplitude of the motion. The second term is of the order of G times the free surface height and is therefore of the order of the amplitude squared. To be consistent with our linearization, such quadratic terms must be neglected. That implies that each term in the boundary con- condition stated above can be applied at z = 0; thus, we have d

at z = 0 C.28) dt oz p dt as the boundary condition on the upper surface, while again at the lower surface, d